cos(a+b)cos(a-b)-sin(a+b)sin(a-b)

Using the formula for the cosine of the difference of two angles and the formula for the sine of the sum of two angles, we can simplify the expression:

cos(a+b)cos(a-b)-sin(a+b)sin(a-b)
= (cos(a)cos(b) - sin(a)sin(b))(cos(a)cos(b) + sin(a)sin(b)) - (sin(a)cos(b) + cos(a)sin(b))(cos(a)sin(b) - sin(a)cos(b))
= cos^2(a)cos^2(b) - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^2
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^2
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^2
= cos(a)^2cos(b)^2 + sin(a)^2sin(b)^2 - sin(a)^2cos(b)^2 - cos(a)^2sin(b)^2
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= cos^2(a)cos^2(b) + sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1 - sin^2(a)sin^2(b) - sin^2(a)cos^2(b) - cos^2(a)sin^2(b)
= 1

Therefore, the simplified expression is 1.